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Statistical Methods for Fuzzy Data by Reinhard Viertl

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16.2 Fuzzy HPD-regions

Classical HPD-regions ΘH, 1−α are defined for standard a posteriori densities π(·|) in the following way:

ΘH, 1−α is a subset of Θ for which

(16.1) Numbered Display Equation

such that

(16.2) Numbered Display Equation

where C is the maximal possible constant such that (16.1) is fulfilled.

In the case of classical a priori distribution and fuzzy data x1*,…,xn* with combined fuzzy sample , the concept of highest posterior density regions can be generalized in the following way:

For confidence level 1−α and all θ ∈ Θ, and ∈ supp(*) = supp[ξ(.,…,.)], a classical subset B(θ, ) of Θ is defined by

Unnumbered Display Equation

The generalized HPD-region for θ based on the fuzzy sample * is the fuzzy subset ΘH, 1−α* whose membership function ψ(·) is determined ...

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